If `3^x-3^(x-2)-72=0`, then the value of `x^(x/2)` is:

Options

A. 9
B. 27
C. 16
D. 4

Answer

To solve the equation \( 3^x - 3^{x-2} - 72 = 0 \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 3^x - 3^{x-2} - 72 = 0 \]
We can rewrite \( 3^{x-2} \) as \( \frac{3^x}{3^2} = \frac{3^x}{9} \). Thus, the equation becomes:
\[ 3^x - \frac{3^x}{9} - 72 = 0 \]

### Step 2: Factor out \( 3^x \)
Next, we can factor out \( 3^x \):
\[ 3^x \left(1 - \frac{1}{9}\right) - 72 = 0 \]
Calculating \( 1 - \frac{1}{9} \) gives us:
\[ 1 - \frac{1}{9} = \frac{9}{9} - \frac{1}{9} = \frac{8}{9} \]
So the equation becomes:
\[ 3^x \cdot \frac{8}{9} - 72 = 0 \]

### Step 3: Isolate \( 3^x \)
Now, we can isolate \( 3^x \):
\[ 3^x \cdot \frac{8}{9} = 72 \]
Multiplying both sides by \( \frac{9}{8} \) gives:
\[ 3^x = 72 \cdot \frac{9}{8} \]

### Step 4: Simplify the right side
Calculating \( 72 \cdot \frac{9}{8} \):
\[ 72 \cdot \frac{9}{8} = \frac{72 \cdot 9}{8} = \frac{648}{8} = 81 \]
Thus, we have:
\[ 3^x = 81 \]

### Step 5: Solve for \( x \)
Since \( 81 = 3^4 \), we can equate the exponents:
\[ x = 4 \]

### Step 6: Find \( x^{(x/2)} \)
Now we need to find \( x^{(x/2)} \):
\[ x^{(x/2)} = 4^{(4/2)} = 4^2 = 16 \]

### Final Answer
Thus, the value of \( x^{(x/2)} \) is:
\[ \boxed{16} \]